Problem 93 - Precalculus : Building Concepts and Connections

Question

In this set of exercises, you will use right triangle trigonometry to study real-world problems. Unless otherwise indicated, round answers to four decimal places. A garden consists of four sections of equal dimensions. Each section is in the shape of a right triangle. One of the acute angles in each triangle measures \(24^{\circ},\) and the side adjacent to that angle is 16 feet long. (IMAGES CANNOT COPY) (a) Find the length of the hypotenuse. (b) Find the length of the side opposite the \(24^{\circ}\) angle. (c) Compute the total area of the garden (all four sections combined).

Step-by-Step Solution

Answer

The length of the hypotenuse is the result of \( \frac{16}{\cos(24^{\circ})} \), the length of the side opposite the angle is the result of \( length \ of \ hypotenuse \times \sin(24^{\circ}) \), and the total area of the garden is 4 times the result of \( \frac{1}{2} \times 16 \ feet \times length \ of \ side \ opposite \ the \ angle \).

Step 1: Find the Hypotenuse Length

First calculate the hypotenuse length using the cosine of the given angle. The cosine of an angle in a right triangle is equal to the adjacent side divided by the hypotenuse. Therefore, the hypotenuse length \( h \) can be found using the formula \( h = \frac{adjacent}{\cos(\theta)} \), where \( \theta \) is the given angle. Plugging in the given values, \( h = \frac{16}{\cos(24)} \) can be calculated.

Step 2: Calculate the Length of the Side Opposite the Angle

For the second step, the sine of the given angle is equal to the length of the side opposite the angle divided by the hypotenuse. Rearranging the equation to solve for the opposite side (\( o \)), the formula \( o = h \cdot \sin(\theta) \) is derived. Given \( h \) from step 1 and \( \theta = 24 \) degrees, \( o = h \cdot \sin(24^{\circ}) \) is the equation to use.

Step 3: Compute the Garden's Total Area

Once the dimensions of one triangle section are known, the area of the garden can be found by calculating the area of one triangle and then multiplying by 4. The area of a triangle is given by the formula \( area = \frac{1}{2} \times base \times height \). In this case, the base is the side adjacent to the given angle and the height is the side opposite the given angle. Following, the area of the garden is 4 times the area of one triangle, use the formula \( total \ area = 4 \times \frac{1}{2} \times adjacent \times o \), where \( o \) is the length of the side opposite the angle from step 2.

Key Concepts

Hypotenuse Length Calculation

Understanding the hypotenuse length calculation in a right triangle is integral to harnessing the power of trigonometry in real-life applications. In our scenario with the garden shaped as right-angled triangles, the hypotenuse represents the side opposite the right angle. To find this length, we utilize the cosine function, which is defined as the ratio of the adjacent side to the hypotenuse in a right triangle.

To compute the hypotenuse when given one angle and the length of the adjacent side, the formula is rearranged from the basic definition of cosine: \( \cos(\theta) = \frac{adjacent}{hypotenuse} \). Solving for the hypotenuse, \( hypotenuse = \frac{adjacent}{\cos(\theta)} \). By substituting the given values, we arrive at the hypotenuse for one section of the garden. It is essential to round the result to four decimal places as per the instructions, ensuring precision in the real-world context.

Sine and Cosine Functions

The sine and cosine functions are fundamental components of right triangle trigonometry. They relate the angles of a triangle to the lengths of its sides. In a right-angled triangle,

• The sine of an angle (\(\sin(\theta)\)) is the ratio of the length of the side opposite the angle to the length of the hypotenuse.
• The cosine of an angle (\(\cos(\theta)\)) is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.

For the garden problem, we have an angle of \(24^\circ\) and the length of the adjacent side. After finding the hypotenuse using the cosine function, we can calculate the length of the side opposite the angle with the sine function. Here, the formula \( o = h \cdot \sin(\theta) \) directs us to multiply the hypotenuse by the sine of the given angle. This step is pivotal in providing the dimensions necessary for later calculations, such as determining the area of the garden.

Area of a Triangle

The area of a triangle is an essential concept in geometry and has practical implications in various fields. The standard formula for the area of a triangle is \( area = \frac{1}{2} \times base \times height \). This equation illustrates that any side of a triangle can be considered the 'base', and the 'height' is the perpendicular distance from the base to the opposite vertex.

Applying this to our quadruple-sectioned garden, once we have the dimensions of a single triangular section, namely the base (adjacent side) and height (opposite side), we can calculate the area of one triangular section. Furthermore, to find the garden's total area, we quintuple the area of a single section because the garden comprises four identical triangular sections. This calculation provides us with a practical approach to measuring spaces accurately and effectively in real-world contexts such as garden planning.